| Abstract: |
Prime-like integer sequences (PLISs) constitute a fundamental class of objects in analytic number theory, exhibiting distributional and structural properties that closely approximate, and in several measurable respects mirror, the behavior of the classical prime number sequence. This empirical study presents a systematic quantitative analysis of four canonical prime-like sequences Lucky Numbers, Semi-primes, Chen Primes, and Quasi-primes benchmarked against the distribution of ordinary primes within the integer interval [1, 10,000]. Employing a hybrid methodology that integrates computational sieve operations, statistical gap analysis, and correlation-based structural profiling, the study measures density ratios, average and variance of inter-term gaps, sieve-step overlaps, and distributional convergence indices across these sequence families. The results confirm a statistically significant structural equivalence between Lucky Numbers and prime numbers, with a Pearson correlation coefficient of r = 0.975 in the range N = 1,000, declining gradually to r = 0.968 at N = 10,000, indicating robust asymptotic alignment. Semi-primes demonstrate the highest density but the lowest structural similarity, while Chen Primes exhibit gap variance profiles closest to the prime number sequence. The data further validate a generalized Goldbach-type pairing hypothesis across PLIS families with a 97.4% overlap in pairwise representations. These findings reinforce the theoretical hypothesis that primality is not an isolated arithmetic property but the statistical apex of a broader hierarchy of multiplicative sieve-based integer families. The study contributes five original empirical data tables and a comparative framework that may inform future research into the Riemann Hypothesis, twin prime conjecture, and broader analytic number theory investigations. |
